#201231

order	512
nilpotency class	3
isoclinism	isoclinic class of #201094
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1043 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,6,10,7,9,5,8)(3,11,12,4,14,15,13,16)(17,18)(19,28)(20,33,36,25)(21,26)(22,37)(23,27)(24,31)(29,42,46,48)(30,39)(32,43)(34,41,35,40)(38,45,47,44);(1,3)(2,4)(5,12)(6,13)(7,14)(8,15)(9,16)(10,11)(17,19,21,30,23,32,22,31)(18,24,37,43,27,39,26,28)(20,29,35,45,36,46,34,44)(25,38,40,48,33,47,41,42);(1,4)(2,3)(5,15)(6,11)(7,16)(8,12)(9,14)(10,13)(17,20)(18,25)(19,29)(21,35)(22,34)(23,36)(24,38)(26,41)(27,33)(28,42)(30,45)(31,44)(32,46)(37,40)(39,47)(43,48);(1,5,7,6)(2,8,9,10)(3,12,14,13)(4,15,16,11)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,34,36,35)(24,28,39,43)(25,40,33,41)(29,44,46,45)(38,48,47,42)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,e^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108427);